Universal relations for nonlinear electroelastic solids

Summary Electro-sensitive elastomers are materials that can support large elastic deformations under the influence of an electric field. There has been growing interest recently in their applications as so-called ``smart materials'. This paper is devoted to the derivation of universal relations in the context of the nonlinear theory of electroelasticity that underpins such applications. Universal relations are equations relating the components of the stress, the electric variables and the deformation that are independent of the constitutive law for a family of materials. For the general constitutive equations of an isotropic electroelastic material derived from a free energy function and for some special cases of these equations, we obtain universal relations, the word ``universal' being relative to the considered class or subclass of constitutive laws. These universal relations are then applied to some controllable states (homogeneous and non-homogeneous) in order to highlight some examples that may be useful from the point of view of experimental characterization of the material properties. Additionally, we examine the (non-controllable) problem of helical shear of a circular cylindrical tube in the presence of a radial electric field, and we find that a nonlinear universal relation that has been obtained previously for an elastic material also holds when the electric field is applied.     

Stability of morphotropic (110) oriented 0.65PMN-0.35PT single crystals

Electromechanical properties of (1-x)Pb (Mg1/3Nb2/3)O3-xPbTiO3 (PMN-PT) single crystals with x = 0.35 were investigated as a function of different external disturbances. The polarization dependence on the electromechanical properties was first studied in order to determine the best polarization path. The correlation with X-ray measured phase ratio is presented and shows that the maximum of electromechanical properties may be correlated with a minimum rhombohedral/tetragonal phase ratio. Temperature, stress, electric field, and time (aging) stability was studied in order to determine performance-limiting factors of these materials. The rhombohedral/tetragonal phase transition is observed on temperature (80 degrees C), inducing a decrease of the electromechanical coupling factor (from 85% to 50%); but the whole properties are recovered while returning to room temperature. Stress measurement shows a large depoling of sample for stresses above 30 MPa. The PMN-PT single crystals were found to be surprisingly stable during aging, except for mechanical and dielectric losses. The same tendency was found on alternating current (AC) electric field dependence.     

The use of stress sweep tests for asphalt mixtures nonlinear viscoelastic and fatigue damage responses identification

Asphaltic materials are known to present a behavior that can be approximated by the theory of viscoelasticity. For these materials it is essential to characterize fatigue damage. An important aspect therein is the separation between nonlinear viscoelastic and fatigue damage responses. This is a complex issue, since both nonlinearity and damage have a similar effect on the overall material mechanical behavior, i.e. decrease in the stiffness and increase in the phase angle. This paper presents an experimental and a mathematical procedure to separate the nonlinear viscoelastic from the fatigue damage response for asphaltic materials. Stress sweep tests were used to characterize a hot mixture asphalt at nine conditions (three temperatures and three frequencies). Once all strain values were obtained in a stress controlled sweep test, a statistical analysis was used to find the maximum stress that can be applied to the material without invoking the damage response. The results showed that the transition stress value is directly associated with material properties, the stiffness being an important factor in this result. Consequently, stress, temperature and frequency determine together the mechanical response of the material (linear or nonlinear viscoelastic, fatigue damage and/or plastic deformation). Results from this study can be associated with other fatigue damage approaches in order to better select the stress or strain amplitude that should be used in fatigue tests, and to eliminate the amount of energy that is dissipated in the nonlinear viscoelastic region.     

力、电、热作用下晶内微裂纹的演化

基于表面扩散的经典理论及其弱解描述,对曲率、力、电和热共同作用下金属材料内部晶内微裂纹的演化进行了有限元分析。详细讨论了微裂纹初始形态、电场大小、应力大小和电致生热对微裂纹演化的影响。结果表明:对于形态比为的微裂纹,存在一临界电场值和临界应力值。当且时,微裂纹逐渐圆柱化;当或时,微裂纹分节为上、下或左、右两个小裂纹。热应力可减小的值,即有利于微裂纹分节。同时热应力可加快微裂纹的漂移速度,缩短分节时间。     

Modeling of columnar recombination for high-energy photon generated electrons and holes in amorphous selenium

An analytical model is developed to study the mechanisms of X-ray generated free Electron–hole pair (EHP) creation energy in amorphous selenium (a-Se) at high electric fields. The model is presented to show the electric field and temperature dependence of the charge extraction yield limited by the columnar recombination for the materials that have widely unequal drift mobility for electrons and holes, such as a-Se. The model is compared with Jaffe’s columnar recombination model with widely varying field and temperature. In addition, the free EHP creation energy is calculated by incorporating the initial charge extraction yield and the charge collection efficacy of the free carriers. Also, the results of this model are compared with the recently published experimental results on EHP creation energy. The analysis of the results confirm that the proposed model gives the best possible explanation to the columnar recombination mechanisms in a-Se and the free EHP creation mechanisms at diagnostic X-ray exposures can be described by the columnar recombination.     

A new nonlinear constitutive theory for conducting magnetothermoelastic solids

A new nonlinear theory of constitutive equations for electrically and thermally conducting magnetothermoelastic (MTE) solids is developed. In the theory, the electric current and heat flux vectors are also considered to be independent variables in the argument of each constitutive function. It is shown that the modified Helmholtz free energy (MH FE) density, which is a thermodynamical potential for the specific entropy, the magnetization and the stress tensor, does no longer appear as a function of the temperature, the magnetic field and the strain tensor, but it also depends upon the electric current and heat flux vectors. Furthermore, referring to the mentioned constitutive equations, the Gibbs equation is also generalized. In order to expose the constitutive theory developed here, an appropriate polynomial expression of the MH FE density for the anisotropic materials is proposed, and, exploiting the method of the theory of invariants, its exact expression is also determined. With the use of these two expressions, a set of rather general nonlinear constitutive equations, which governs a lot of magnetoelastothermo-electrical (MET-E) effects, is then obtained explicitly. It is interesting to notice that each of the constitutive equations mentioned above has a pseudo (ir) reversible part in vicinities of the new equilibrium state, namely the thermo-electrical equilibrium (T-EE) state. According to the deductive scheme, the generalized constitutive equations and the Gibbs equation in the present work are finally discussed for special materials, and/or vanishing some of the fields. The resulting expressions are, as they should be, in full mutual agreement with the established theories on the same subject.     

Variational‐based modeling of micro‐electro‐elasticity with electric field‐driven and stress‐driven domain evolutions

Recently, increasing interest in so‐called functional or smart materials with electromechanical coupling has been shown such as ferroelectric piezoceramics. These materials are characterized by microstructural properties, which can be changed by external stress and electric field stimuli, and hence find use as the active components in sensors and actuators. The electromechanical coupling effects result from the existence and rearrangement of microstructural domains with uniformly oriented electric polarization. The understanding and efficient simulation of these highly nonlinear and dissipative mechanisms, which occur on the microscale of ferroelectric piezoceramics, are a key challenge of the current research. This paper does not offer a substantially new physical model of these phenomena but a new mathematical modeling approach based on a rigorous exploitation of rate‐type variational principles. This provides a new insight in the structure of the coupled problem, where the governing field equations appear as the Euler equations of a variational statement. We outline a variational‐based micro‐electro‐elastic model for the microstructural evolution of both electrically and mechanically driven electric domains in ferroelectric ceramics, which also incorporates the surrounding free space. To this end, we extend recently developed multifield incremental variational principles of electromechanics from local to gradient‐extended dissipative response and specialize it by a Ginzburg–Landau‐type phase field model, where the thickness of the domain walls enters the formulation as a length scale. This serves as a natural starting point for a canonical compact, symmetric finite element implementation, considering the mechanical displacement, the microscopic polarization, and the electric potential induced by the polarization as the primary fields. The latter is defined on both the solid domain and a surrounding free space. Numerical simulations treat domain wall motions for electric field‐driven and stress‐driven loading processes, including the expansion of the electric potential into the free space. Copyright © 2012 John Wiley & Sons, Ltd.     

Normally distributed free energy model and creep behavior of ferroelectric polycrystals at room and high temperatures

A constitutive model that can be used to predict creep behavior of ferroelectric polycrystals at room and high temperatures is proposed. The model consists of the Gibbs free energy function with normal distribution and a switching evolution law with critical driving force. Linear moduli in the free energy function and switching parameters in the switching law are assumed to be linearly dependent on temperature. A ferroelectric polycrystal is modeled by an agglomerate of 210 single crystallites. Compressive stress and electric field-induced creep behavior as well as polarization hysteresis and strain butterfly responses of the model are calculated and compared with experimental observations.     

A radial point interpolation meshless method extended with an elastic rate-independent continuum damage model for concrete materials

An advanced discretization meshless technique, the radial point interpolation method (RPIM), is applied to analyze concrete structures using an elastic continuum damage constitutive model. Here, the theoretical basis of the material model and the computational procedure are fully presented. The plane stress meshless formulation is extended to a rate-independent damage criterion, where both compressive and tensile damage evolutions are established based on a Helmholtz free energy function. Within the return-mapping damage algorithm, the required variable fields, such as the damage variables and the displacement field, are obtained. This study uses the Newton–Raphson nonlinear solution algorithm to achieve the nonlinear damage solution. The verification, where the performance is assessed, of the proposed model is demonstrated by relevant numerical examples available in the literature.     

Improvement of aging characteristics of (Mn, Nb, Er)-doped ZnO–V2O5-based varistor ceramics by small sintering changes

This paper focuses on that the aging characteristics for (Mn, Nb, Er)-doped ZnO–V₂O₅-based varistor ceramics could be controlled by small sintering change. Small change in the sintering temperature did have a noticeable effect on the DC aging characteristics as well as initial electric field-current density (E–J) characteristics. As the sintering temperature increased, the average grain size increased from 4.3 to 9.0 μm, whereas the sintered density decreased from 5.57 to 5.47 g/cm³. The breakdown field and the nonlinear coefficient decreased from 7,408 to 1,424 V/cm and from 55.0 to 24.1, respectively, with an increase in the sintering temperature. Concerning stability, the varistor ceramics sintered at 925 °C exhibited surprisingly stable accelerated aging characteristics, with %ΔE_{1 mA} = 1.4 % and %Δα = 13 % for DC accelerated aging stress of 0.85 E_{1 mA}/85 °C/24 h.     

A 3D Constitutive Model for Magnetostrictive Materials

This paper is concerned with a 3-D general constitutive law of nonlinear magneto-thermo-elastic coupling for magnetostrictive materials. The model considered here is thermodynamically motivated and based on the Gibbs free energy function. A set of closed and analytical expressions of the constitutive relationships for the magnetostrictive materials are obtained, in which all parameters can be determined by those measurable experiments in mechanics and physics. Then the model can be simplified to two cases, i.e. magnetostrictive rods and films. It is found that the predictions from this model are in good accordance with the experimental data including both rods and films. In particular, the effects of pre-stress or in-plane residual stress and environment temperature on the magnetization or the magnetostriction are also discussed.     

纳米晶材料在外场作用下的结构稳定性研究

综述了近年来纳米晶材料在外场作用下的结构稳定性的研究进展,着重介绍了纳米晶材料在温度场和应力场、电场、磁场联合作用下的晶粒长大行为。对纳米晶材料而言,外加应力场促进其晶粒长大;外加电场抑制其晶粒长大;而外加磁场的影响则有待进一步研究。     

Effect of MWCNTs/ZnO inorganic fillers on the electrical,mechanical and thermal properties of SiR-based composites

Rubber insulation materials were widely used in the fields of electrical and electronic engineering, especially, which have excellent nonlinear electrical conductivity and can be employed to homogenize the electric field distribution of cable accessories. To enable the rubber materials, such as silicon rubber (SiR), to possess excellent nonlinear electrical conductivity has been a hot issue. In this paper, MWCNTs/ZnO inorganic fillers were prepared by mixing a small amount of multi-wall carbon nanotubes (MWCNTs) with zinc oxide (ZnO) nanosheets, and MWCNTs/ZnO/SiR composites were prepared. The macroscopical properties results show that the nonlinear electrical conductivity characteristics can be induced by filling appropriate content of MWCNTs/ZnO fillers, and the threshold field strength corresponding to the nonlinear conductivity gradually decreases with the increase of MWCNTs filling content, which further decreases with the increase of measured temperature. The COMSOL simulation results also verify that MWCNTs/ZnO/SiR composite with nonlinear conductivity can effectively reduce the electric field strength at the stress cone of cable accessories. In addition, the thermal conductivity and tensile strength for MWCNTs/ZnO/SiR composite are also improved comparing to pristine SiR. This work demonstrates MWCNTs/ZnO/SiR composites possess outstanding overall properties and have good potential to be used in the cable accessory.

     

A computational model for the finite element analysis of thermoplasticity coupled with ductile damage at finite strains

An updated Lagrangian implicit FEM model for the analysis of large thermo‐mechanically coupled hyperelastic‐viscoplastic deformations of isotropic porous materials is considered. An appropriate framework for constitutive modelling is introduced that includes a stress‐free thermally expanded configuration and a plastically deformed unstressed damaged configuration. A two‐level iterative scheme is employed at each time increment to solve the field equations governing the conservation of momentum (mechanical step) and the conservation of energy (thermal step) for the coupled thermo‐mechanical problem. Exact linearizations for the calculation of the tangent stiffness are performed in each of these solution steps. A fully implicit, thermo‐mechanically coupled and incrementally objective Euler‐backward radial return based map is developed for the time integration of the constitutive equations. The present model is used to analyse a number of benchmark examples including metal forming processes wherein temperature and the accumulated damage play an important role in influencing the deformation mechanism and the nature of the deformed workpiece. Copyright © 1999 John Wiley & Sons, Ltd.     

Nonlinear effects in the kinetics of fracture of polymers and composites

A generalization of the well-known Zhukov formula is proposed to describe the temperature-time dependence of strength in uniaxial tension. The exponent in the Zhukov formula is represented in the form of a segment of a series expansion of free energy of activation as a function of stress and temperature. Only the term which is linear with respect to stress is retained in the series. It is shown that other terms must be kept in the series in order to explain certain deviations from the simplest linear Zhukov model. Thus, the addition of a term which is linear with respect to temperature yields the well-known Zhukov-Bartenev formula, keeping the nonlinear cross term yields the Zhukov-Ratner formula, etc. The free energy of activation can be expanded into a series because it is an analytic function of stress and temperature at the zero point (which is not always the case). In the case of the failure of elastomers, free energy has a logarithmic singularity in relation to stress and thus cannot be expanded into a series.Translated from Problemy Prochnosti, No. 6, pp. 13–17, June, 1993.We are deeply to G. M. Bartenev for his advice and valuable comments regarding our study.     

Dynamic fracture mechanics study of an electrically impermeable mode III crack in a transversely isotropic piezoelectric material under pure electric load

The dynamic response of an electrically impermeable Mode III crack in a transversely isotropic piezoelectric material under pure electric load is investigated by treating the electric loading process as a transient impact load, which may be more appropriate to mimic the real service environment of piezoelectric materials. The stress intensity factor, the mechanical energy release rate, and the total energy release rate are derived and expressed as a function of time for a given applied electric load. The theoretical results indicate that a purely electric load can fracture the piezoelectric material if the stress intensity factor or the mechanical energy release rate is used as a failure criterion.     

Effect of inertia on electrified film flow over a wavy wall

The effect of inertia on the steady flow of a liquid layer down a wavy wall in the presence of an electric field is investigated. Both the liquid film and the region above it are assumed to act as perfect dielectrics. A linearised perturbation analysis is performed for flow down a wall with small-amplitude sinusoidal corrugations, and the free-surface amplitude and phase shift are computed numerically for a broad range of flow conditions. It is shown that the electric field can be used to manipulate the phase shift between the free surface and the wall. In particular, when the Reynolds number lies below a threshold value, an electric field of sufficient strength will bring the free surface precisely into phase with the wall. An electric field can also be used to mitigate the resonance effect identified by previous workers, in which the free surface suffers significant amplification in comparison to the height of the wall corrugations at a particular Reynolds number. Working on the basis of the lubrication approximation, a nonlinear equation for the film thickness is derived featuring a non-local term due to the electric field. Numerical solutions for flow over a wavy wall of finite amplitude reveal that the effect of inertia on the free-surface characteristics depends on the electrical properties of the fluid layer and the strength of the imposed electric field.     

Energy-based mechanistic approach for damage characterization of pre-flawed visco-elasto-plastic materials

Damage characterization plays a significant role in producing durable and high performance structural materials. However, it is somewhat complicated because of the particular characteristics of many materials, such as pre-existing flaws, time-dependent behaviors, and coexistence of cracking and permanent deformation. This kind of materials is pre-flawed visco-elasto-plastic material. In order to characterize damage in such materials, this paper proposes an energy-based mechanistic (EBM) approach that provides a complete solution to these problems. As typical pre-flawed visco-elasto-plastic materials, asphalt mixtures are selected to demonstrate the principles and applications of the EBM approach.When an asphalt mixture is not damaged, the pre-existing flaws are air voids, characterized by the average air voids size and number of air voids calculated by the EBM approach. The calculated values are more accurate than those measured by the X-ray Computed Tomography system. Due to the increased accuracy, it is discovered that the air voids becomes smaller when the mixture is aged, which serves as an evidence of the change of the internal structure of the material due to aging.When an asphalt mixture is damaged, the damage includes cracking and permanent deformation. The cracking damage is a multitude of randomly distributed cracks. A new concept, distributed continuum fracture (DCF), is introduced to model the distributed cracks in the EBM approach. Development of cracking damage is quantified by the evolution of damage density, average crack size and number of cracks. The damage densities of eight different mixtures are proven to correctly reflect the effect of mixture composition and aging. New features of number of cracks discovered lead to new definitions of cracking history in pre-flawed materials. The energy for permanent deformation is separated from that expended for cracking in the same asphalt mixture. Such a separation acknowledges the fact that cracking and permanent deformation always occur simultaneously. The separated energy for cracking is used to define a cracking energy dissipation rate, a direct indicator of cracking susceptibility of asphalt mixtures.In a word, the EBM approach is able to characterize damage in asphalt mixtures under various conditions using one type of test on one specimen. It requires simple inputs: stress, strain, and time, and all the calculations are performed automatically by the Excel. Using this approach to analyze the test data is more efficient than some alternative methods because of less testing effort and more informative results with improved accuracy.     

Quantitative characterization of accelerated aging in cement composites using flexural inverse analysis

A constitutive model consisting of a tri-linear tensile stress-strain with residual strength was applied in characterization and prediction of long term flexural behavior of several cement-based composite materials. Flexural test results were back-calculated to obtain material parameters and establish their relationship with aging. The material behavior is described by tensile stress-strain parameters consisting of elastic modulus, first cracking strain, post cracking stiffness, ultimate strain, and a residual strength parameter. The relationships between the material parameters and age were established by studying the time dependent flexural performance of various composites with glass and natural fibers as reported by Litherland et al. (1981), Marikunte et al. (1997), Bartos et al. (1996), and natural fibers reported by Toledo-Filho et al. (2000). An analytical model for prediction of rate and extent of damage as a function of time and temperature is proposed for degradation of flexural behavior of strain softening and hardening fiber reinforced concrete subjected to aging. This model is applicable to long-term durability of different classes of materials subject to accelerated aging under different environmental conditions.     

Nonlinear Viscoelastic and Viscoplastic Constitutive Equations Based on Thermodynamics

An approach to modeling the mechanical behavior of fiber reinforced and unreinforced plastics with an evolving internal state is described. Intrinsic nonlinear viscoelastic and viscoplastic behavior of the resin matrix is taken into account along with growth of damage. The thermodynamic framework of the method is discussed first. The Gibbs free energy is expressed in terms of stresses, internal state variables (ISVs), temperatureand moisture content. Simplifications are introduced based on physical models for evolution of the ISVs and on experimental observations of thedependence of strain state on stress state and its history. These simplifications include use of master creep functions that account for multiaxial stresses, environmental factors and aging in a reduced time and other scalars. An explicit representation of the strains follows, which isthen specialized to provide three-dimensional homogenized constitutiveequations for transversely isotropic, fiber composites. Experimentalsupport for these equations is briefly reviewed. Finally, physicalinterpretation of some of the constitutive functions is discussed usingresults from a microcracking model as well as molecular rate process andfree volume theories. It is shown that the present thermodynamicformulation leads to a generalized rate process theory that accounts for abroad distribution of thermally activated transformations in polymers.